The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Trujić showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the upper bound of $n^{5/3 + o(1)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.